Anna BAHYRYCZMagdalena PISZCZEK

Institute of Mathematics，Pedagogical University，Podchor?a˙zych 2，30-084 Krak′ow，Poland

E-mail∶bah@up.krakow.pl；magdap@up.krakow.pl

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ON APPROXIMATELY（p，q）-WRIGHT AFFINE FUNCTIONS AND INNER PRODUCT SPACES?

Anna BAHYRYCZMagdalena PISZCZEK

Institute of Mathematics，Pedagogical University，Podchor?a˙zych 2，30-084 Krak′ow，Poland

E-mail∶bah@up.krakow.pl；magdap@up.krakow.pl

We prove，using the fixed point approach，some results on hyperstability（in normed spaces）of the equation that defines the generalization of p-Wright affine functions and show that they yield a simple characterization of the complex inner product spaces.

Hyperstability；p-Wright affine function；complex inner product space；fixed point theorem

2010 MR Subject Classification39B52；39B55；47H10

1　Introduction

Let X and Y be linear spaces over fields F∈｛R，C｝and K，respectively，and p，q∈F be fixed.The functional equation

for function g：X→Y，generalizes the equation

For X=Y=R and p∈（0，1），the solutions of equation（1.2）are called the p-Wright affine functions.The functions satisfying equation（1.1）are called the（p，q）-Wright affine functions.

The definition of p-Wright affine functions is connected to the notions of p-Wright convexity and p-Wright concavity（see，for example，［9，10，12，14，17］）.Clearly，for p=1/2，equation（1.2）is just the well known Jensen's equation

For p=1/3，equation（1.2）takes the form

which was investigated by Najati and Park［13］；in particular，they proved some results on its stability and applied them in the investigation of the generalized（σ，τ）-Jordan derivations on Banach algebras.

In this article，we present some hyperstability results for equation（1.1）.Namely，we prove that for some particular forms of θ，the functional equation（1.1）is θ-hyperstable in the class of functions g：X→Y，that is，each g：X→Y satisfying the inequality

‖g（px+qy）+g（qx+py）?g（x）?g（y）‖≤θ（x，y），x，y∈X，

must be（p，q）-Wright affine function（see Theorems 3.1 and 3.2）.We show that they yield a simple characterization of complex inner product spaces（see Corollary 4.1）.For

with some α∈R，equation（1.1）characterizes norms in the complex inner product spaces.

It is known（see［16］）that every solution g of（1.1）has the form

with some c∈Y，an additive A：X→Y，and a quadratic B：X→Y.

For more information and numerous on the equation of the p-Wright affine functions and its stability，please refer to，for example，［2-4，9，11，15］.

2　Auxiliary Results

To present an auxiliary（fixed point）result，we need to introduce some necessary hypotheses（R+stands for the set of nonnegative reals and ABdenotes the family of all functions mapping a set B 6=?into a set A 6=?）.

（H1）X is a nonempty set，E is a Banach space，f1，···，fk：X→X and L1，···，Lk：X→R+are given，and T：EX→EXis an operator satisfying the inequality

Now，we can present the above mentioned fixed point theorem proved in［6，Theorem 1］（see also［7，Theorem 2］and［5，8］）.

Theorem 2.1Let hypotheses（H1），（H2）be valid and functions ε：X→R+and ?：X→E fulfil the following two conditions：

Then，there exists a unique fixed point ψ of T with

Moreover，

First，we prove an auxiliary lemma，which shows that（p，q）-Wright affine function on X｛0｝is（p，q）-Wright affine function on the whole X.

Lemma 2.2Let p，q∈F｛0｝.If a function g：X→Y satisfies

then g is（p，q）-Wright affine function.

ProofAssume that g fulfils（2.1）.Let ge，go：X→Y denote the even and the odd parts of g，respectively，that is，

Putting y=x in（2.1），we have

Setting y=?x in（2.1），we obtain

Thus，the even part of g satisfies

Replacing in（2.1）x by px，y by?qx，we get

and hence using（2.2）and（2.3），we obtain

Replacing in（2.1）x by px，y by qy，we have

Setting y=x in（2.6），we get

Adding equations（2.4）and（2.7），we obtain，for the odd part of g，

Whence using（2.1），we have，for x∈X｛0｝，

thus

Next，applying（2.6）and（2.2），we obtain

Replacing in the above equation x bywe obtain

which together with（2.5）gives

and finishes the proof.

3　Hyperstability Results

In this section we present the results concerning hyperstability of equation（1.1）.

Theorem 3.1Let X be a normed space over a field F∈｛R，C｝，Y be a normed space，p，q∈F｛0｝，c≥0，k＞0，and l＜0.Then，every function g：X→Y with

satisfies equation（1.1）.

ProofFirst，without loss of generality，we assume that Y is a Banach space，because otherwise we can replace it by its completion.

Replacing y by mx in（3.1），for m∈N，we get

Write

and

Define

so（H1）is valid.

Next，we can find m0∈N，such that

Therefore，

for m≥m0and x∈X｛0｝.

Thus，according to Theorem 2.1，for each m≥m0，there exists a unique solution Gm：X｛0｝→Y of the equation

such that

for x∈X｛0｝.Moreover，

Now，we show that

for every x，y∈X｛0｝，n∈N0（nonnegative integers）.

If n=0，then（3.3）is simply（3.1）.So，take i∈N0and suppose that（3.3）holds for n=i and x，y∈X｛0｝.Then，

Letting n→∞in（3.3），we obtain

for x∈X｛0｝.Hence，letting m→∞，（2.1）holds.So，according to Lemma 2.2 the function g is the（p，q）-Wright affine function，which completes the proof.

In the analogous way，we can prove the following theorem.

Theorem 3.2Let X be a normed space over a field F∈｛R，C｝，Y be a normed space，c≥0，k＞0，l＜0，and p，q∈F｛0｝，with q 6=?p.Then，every function g：X→Y with

satisfies equation（1.1）.

We notice that for p=q，from（3.5）we obtain

For x∈X｛0｝，we define

Then，（3.6）（if p=q）and（3.5）（if p 6=q）take form

and it is easily seen that Λmthat has the form described in（H2）and（H1）is valid.Write

Next，we can find m0∈N，such that αm＜1 for all m≥m0.Therefore，after simple calculations，we have

Hence，according to Theorem 2.1，for each m∈Nm0，there exists a unique fixed point Gm：X｛0｝→Y of Tmwith

Moreover，

It is easy to prove that for every x，y∈X｛0｝，m∈Nm0，and n∈N0，

Letting n→∞in（3.8），we obtain

Remark 3.3We notice that in the particular case when q=1?p，we obtain some results for hyperstability of p-Wright affine functions. The following example show that the assumption q 6=?p in the above theorem is essential. Example 3.4Let g：R→R be defined as g（x）=x2k+1for x 6=0 and g（0）=0，where k is a negative integer number.Then，g satisfies

but g does not satisfy（p，?p）-Wright affine equation，because g is not a constant.

4　Characterization of the Complex Inner Product Spaces

In this part，we show that Theorem 3.1 and Theorem 3.2 yield a characterization of the complex inner product spaces.

First，we define the function θj（x，y）：（X｛0｝）×（X｛0｝）→R+，for j∈｛1，2｝in the following way

where d≥0，k＞0，l＜0.

Corollary 4.1The following three statements are valid.

（i）Let X be a normed space over F∈｛R，C｝.Then，for every s＞0，j∈｛1，2｝and p，q∈F｛0｝with|p+q|6=1 or|p?q|6=1，and additionaly q 6=?p when j=2，we have

（ii）Assume that X is the complex normed space and there exist j∈｛1，2｝，p，q∈CR with q 6=?p when j=2，such that

Then，X is an inner product space and|p+q|=|p?q|=1.

（iii）Let X be an inner product space over F∈｛R，C｝.Then，

ProofTake s＞0，j∈｛1，2｝and p，q∈F｛0｝with|p+q|6=1 or|p?q|6=1，additionaly q 6=?p if j=2 and suppose that

This means that a function g：X→R，g（x）=‖x‖s，satisfies

with some M≥0.Consequently，in view of Theorem 3.1 if j=1，and Theorem 3.2 if j=2，we have

Setting in（4.4）y=x and then y=?x，we get

and

respectively.Hence，|p+q|=|p?q|=1，which is a contraction.

For the proof of（ii），observe that（4.1）is just condition（4.2）with s=2.Hence，

and|p+q|=|p?q|=1，and therefore（1.4）holds with some c∈R，an additive A：X→R，and a quadratic B：X→R.From the fact that g is even and g（0）=0，we obtain

which means that for every x，y∈X，we have the parallelogram equality

and consequently，X is an inner product space.

It remains to show（iii）.So，fix p，q∈F with|p+q|=|p?q|=1.Note that the case F=R is trivial，because then|p|=1 and q=0 or p=0 and|q|=1.So，assume that F=C.Let〈x，y〉denote the inner product of vectors x，y∈X.Write

Then，

whence（with x replaced by px and y by qx）we getNext，by simple calculations，we get Consequently，we obtain

Remark 4.2The condition|p+q|=|p?q|=1 implies that|p|2+|q|2=1.If p，q∈R，then|p|=1 and q=0 or p=0 and q=|1|.Moreover，every pair（p，q）∈C2｛（0，0）｝satisfying the condition|p+q|=|p?q|=1 is of the form（1.3）with some α∈R，and for such pairs（p，q），equation（1.1）characterizes norms in the complex inner product spaces.

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December 4，2014；revised January 8，2015.